Two-dimensional invasion percolation and incipient infinite clusters

Let G=(V,E) be an infinite connected graph in which a distinguished vertex, the origin, is chosen. The edges of G are assigned i.i.d. uniform random variables on [0,1], called weights. The invasion percolation cluster (IPC) of the origin on G is defined as the limit of an increasing sequence (G_n) of connected subgraphs of G as follows. Define G₀ to be the origin. Given G_n = (V_n,E_n), E_n+1 is obtained from E_n by adding to it the edge from the boundary of G_n with the smallest weight. Let G_n+1 be the graph induced by the edge set E_n+1.

Invasion percolation is closely related to critical percolation. In this talk we give some basic relations between invasion percolation and critical percolation. We show how they can be used to study the structure of the two-dimensional IPC. We also show relations to Kesten's incipient infinite cluster and to multiple-armed IICs.

(joint work with Michael Damron and Balint Vagvolgyi)